44 research outputs found
Stability and Hopf-Bifurcation Analysis of Delayed BAM Neural Network under Dynamic Thresholds
In this paper the dynamics of a three neuron model with self-connection and distributed delay under dynamical threshold is investigated. With the help of topological degree theory and Homotopy invariance principle existence and uniqueness of equilibrium point are established. The conditions for which the Hopf-bifurcation occurs at the equilibrium are obtained for the weak kernel of the distributed delay. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and central manifold theorem. Lastly global bifurcation aspect of such periodic solutions is studied. Some numerical simulations for justifying the theoretical analysis are also presented
Synchronous dynamics of a delayed two-coupled oscillator
This paper presents a detailed analysis on the dynamics of a delayed two-coupled oscillator. Linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations
Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays
We consider the van der Pol equation with discrete and distributed delays. Linear
stability of this equation is investigated by analyzing the transcendental characteristic equation of
its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations
by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf
bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis
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Dynamics of neural systems with time delays
Complex networks are ubiquitous in nature. Numerous neurological diseases, such as
Alzheimer's, Parkinson's, epilepsy are caused by the abnormal collective behaviour of
neurons in the brain. In particular, there is a strong evidence that Parkinson's disease is
caused by the synchronisation of neurons, and understanding how and why such synchronisation
occurs will bring scientists closer to the design and implementation of appropriate
control to support desynchronisation required for the normal functioning of the brain. In
order to study the emergence of (de)synchronisation, it is necessary first to understand
how the dynamical behaviour of the system under consideration depends on the changes
in systems parameters. This can be done using a powerful mathematical method, called
bifurcation analysis, which allows one to identify and classify different dynamical regimes,
such as, for example, stable/unstable steady states, Hopf and fold bifurcations, and find
periodic solutions by varying parameters of the nonlinear system.
In real-world systems, interactions between elements do not happen instantaneously
due to a finite time of signal propagation, reaction times of individual elements, etc.
Moreover, time delays are normally non-constant and may vary with time. This means
that it is vital to introduce time delays in any realistic model of neural networks. In
this thesis, I consider four different models. First, in order to analyse the fundamental
properties of neural networks with time-delayed connections, I consider a system of four
coupled nonlinear delay differential equations. This model represents a neural network,
where one subsystem receives a delayed input from another subsystem. The exciting
feature of this model is the combination of both discrete and distributed time delays, where
distributed time delays represent the neural feedback between the two sub-systems, and the
discrete delays describe neural interactions within each of the two subsystems. Stability
properties are investigated for different commonly used distribution kernels, and the results
are compared to the corresponding stability results for networks with no distributed delays.
It is shown how approximations to the boundary of stability region of an equilibrium point
can be obtained analytically for the cases of delta, uniform, and gamma delay distributions.
Numerical techniques are used to investigate stability properties of the fully nonlinear
system and confirm our analytical findings.
In the second part of this thesis, I consider a globally coupled network composed of
active (oscillatory) and inactive (non-oscillatory) oscillators with distributed time delayed
coupling. Analytical conditions for the amplitude death, where the oscillations are quenched,
are obtained in terms of the coupling strength, the ratio of inactive oscillators, the width
of the uniformly distributed delay and the mean time delay for gamma distribution. The
results show that for uniform distribution, by increasing both the width of the delay distribution
and the ratio of inactive oscillators, the amplitude death region increases in the
mean time delay and the coupling strength parameter space. In the case of the gamma
distribution kernel, we find the amplitude death region in the space of the ratio of inactive
oscillators, the mean time delay for gamma distribution, and the coupling strength for
both weak and strong gamma distribution kernels.
Furthermore, I analyse a model of the subthalamic nucleus (STN)-globus palidus (GP)
network with three different transmission delays. A time-shift transformation reduces the
model to a system with two time delays, for which the existence of a unique steady
state is established. Conditions for stability of the steady state are derived in terms of
system parameters and the time delays. Numerical stability analysis is performed using
traceDDE and DDE-BIFTOOL in Matlab to investigate different dynamical regimes in
the STN-GP model, and to obtain critical stability boundaries separating stable (healthy)
and oscillatory (Parkinsonian-like) neural ring. Direct numerical simulations of the fully
nonlinear system are performed to confirm analytical findings, and to illustrate different
dynamical behaviours of the system.
Finally, I consider a ring of n neurons coupled through the discrete and distributed
time delays. I show that the amplitude death occurs in the symmetric (asymmetric) region
depending on the even (odd) number of neurons in the ring neural system. Analytical
conditions for linear stability of the trivial steady state are represented in a parameter space
of the synaptic weight of the self-feedback and the coupling strength between the connected
neurons, as well as in the space of the delayed self-feedback and the coupling strength
between the neurons. It is shown that both Hopf and steady-state bifurcations may occur
when the steady state loses its stability. Stability properties are also investigated for
different commonly used distribution kernels, such as delta function and weak gamma
distributions. Moreover, the obtained analytical results are confirmed by the numerical
simulations of the fully nonlinear system
Fixed-time synchronization problem of coupled delayed discontinuous neural networks via indefinite derivative method
In this brief, we introduce a class of coupled delayed nonautonomous neural networks (CDNNs) with discontinuous activation function. Different from the conventional Lyapunov method, this brief uses the implementation of an indefinite derivative to deal with the nonautonomous system for the case that the topology between neurons is nonlinear coupling, and the system can achieve synchronization in fixed time by selecting the suitable control scheme. The settling time estimation of the system which can get rid of the dependence on the initial value is given. Finally, two examples are given to verify the correctness of the results in this paper
General criteria for asymptotic and exponential stabilities of neural network models with unbounded delays
For a family of differential equations with infinite delay, we give sufficient conditions for the global asymptotic, and global exponential stability of an equilibrium point. This family includes most of the delayed models of neural networks of Cohen-Grossberg type, with both bounded and unbounded distributed delay, for which general asymptotic and exponential stability criteria
are derived. As illustrations, the results are applied to several concrete models studied in the literature, and a comparison of results is given.Fundação para a Ciência e a Tecnologia (FCT) - 2009-ISFL-1-209Universidade do Minho. Centro de Matemática (CMAT
Synchronization analysis of coupled fractional-order neural networks with time-varying delays
In this paper, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems
Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations
This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper